The accurate numerical solution of the Schrödinger equation with an explicitly time-dependent Hamiltonian

Transcription

1 The accurate numerical solution of the Schrödinger equation with an explicitly time-dependent Hamiltonian Marnix Van Daele Department of Applied Mathematics, Computer Science and Statistics Ghent University Congreso Bienal de la Real Sociedad Matemática Española Zaragoza, January 30 - February

2 A reference... This talk is based on V. Ledoux, M. Van Daele Solving the time-dependent Schrödinger problem using the CP approach Computer Physics Communications 185 (2014)

3 Sturm-Liouville problems MATSLISE is a Matlab package for solving Sturm-Liouville and Schrodinger equations It has been developed by Veerle Ledoux under the supervision of Guido Vanden Berghe and VD in a close collaboration with Liviu Ixaru (Bucharest). V. Ledoux, VD Matslise 2.0, a Matlab toolbox for Sturm-Liouville computations, Transactions on Mathematical Software, 22 4 (2016) article 29.

4 Sturm-Liouville problems Find an eigenvalue E and corresponding eigenfunction y(x) 0 such that (p(x)y (x)) + q(x)y(x) = Ew(x)y(x) with boundary conditions a 1 y(a) + a 2 p(a)y (a) = 0, b 1 y(b) + b 2 p(b)y (b) = 0, where a 1 + a 2 0 b 1 + b 2. Schrödinger equation : special case whereby p = w = 1 y + q(x) y = E y

5 Regular Sturm-Liouville problems The eigenvalues of a regular SLP can be ordered as an increasing sequence tending to infinity E 0 < E 1 < E 2 <... E k has index k. The corresponding y k (x) has exactly k zeros on the open interval (a, b). Distinct eigenfunctions are orthogonal with respect to w(x) b a y i (x)y j (x)w(x)dx = 0, i j.

6 How to solve Sturm-Liouville problems Problem: as the index k increases, the corresponding y k become increasingly oscillatory. x y x x 10 4 Standard numerical methods for ODEs encounter difficulties in efficiently estimating the higher eigenvalues. Naive integrators will be forced to take increasingly smaller steps, thereby rendering them exceedingly expensive.

7 Example : the Paine problem y + exp(x) y = E y y(0) = y(π) = 0 The errors ( 10 3 ) obtained with the Numerov algorithm, the Numerov method with correction technique and the EF-Numerov scheme. k E k 10 3 (E k Ẽk) 10 3 (E k ẼCT,k) 10 3 (E k E EF,k )

8 How to solve Sturm-Liouville problems Taking into account the characteristic features of the SL problem, one can construct specialized numerical algorithms having some crucial advantages over general-purpose codes. Some early (seventies to nineties) codes on SL problems : SLEIGN (Bailey et al.) SLEDGE (Fulton-Pruess) SL02F (Marletta-Pryce) SLCPM12 (Ixaru-Vanden Berghe-De Meyer) MATSLISE originates from SLCPM12. In the next slides we focus on the special techniques used in MATSLISE.

9 Basic ideas: 1. Shooting Shooting methods transform the boundary value problem into an initial value problem. One solves the differential equation for a succession of trial values of E which are adjusted till the boundary conditions at both ends can be satisfied at once, at which point we have an eigenvalue. The simplest technique is to shoot from a to b, but multiple shooting is preferred. The eigenvalues are the solutions of a mismatch function φ(e): φ(e) = y L (x m, E)p(x m )y R (x m, E) y R (x m, E)p(x m )y L (x m, E).

10 Basic ideas: 2. Prüfer transformation Suppose we have found an eigenvalue : φ(e) = 0. What is its index? Solution: use the (scaled) Prüfer transformation. 5*pi 4*pi 3*pi 2*pi θ L (x) pi 0 y(x) pi 2*pi θ R (x) 3*pi 4*pi

11 Basic Ideas: 3. Coefficient Approximation Idea behind second order Pruess method: Let a = x 0 < x 1 < x 2 < < x n = b be a partition of [a, b]. Replace (p(x)y (x)) +q(x)y(x) = Ew(x)y(x) in the interval (x i 1, x i ), i = 1,..., n by ( py (x)) + qy(x) = E wy(x). where p, q, w are constant.

12 Basic Ideas: 4. Perturbation theory Higher order methods are obtained by expressing the solution as a series and by adding correction terms to the second order method. All terms in this series can be computed analytically if the coefficient functions are polynomials. Therefor, we first approximate the coefficient functions by polynomials of degree N

13 CPM[N,Q] : Convergence results for problems that are reformulated in Schrödinger form... Let N be the degree of the polynomial approximating q(x) and let Q be the number of correction terms. The corresponding CPM method is denoted is CPM[N,Q]. L.Gr. Ixaru, H. De Meyer and G. Vanden Berghe, CP methods for the Schrödinger equation, revisited, J. Comput. Appl. Math. 88 (1997) The error in the eigenvalue E k, obtained with CPM[N,Q], is of order O(h 2N+2 ) for small E if Q 2 3 N + 1 is of order O(h 2N )/ E for large E if Q δ N0

14 Time-dependent Schrödinger equation ψ(x, t) ı = Ĥ(x, t)ψ(x, t), x R, t > 0 t with the one-dimensional time-dependent Hamiltonian 2 Ĥ(x, t) = 1 + V (x, t), 2µ x 2 initial data ψ(x, 0) = ψ 0 (x) µ is the reduced mass units are chosen such that = 1

15 Standard techniques Standard techniques lead to the linear system ı d Z (t) = H(t)Z (t) dt Z = (Z 1,..., Z N ) T H: N N hermitian matrix

16 Standard technique 1 Discretisation of the spatial variable x Define ψ(x j, t) = Z j (t), j = 1,..., N ı d dt Z j(t) = 1 Z j+1 (t) 2 Z j (t) + Z j 1 (t) 2µ x 2 + V (x j, t)

17 Standard technique 2 Spectral decomposition of ψ(x, t): ψ(x, t) = N Z m (t) y m (x) m=1 whereby {y m (x) m N} is an orthonormal basis (of eigenfunctions of some time-independent Hamiltonian) E.C. Titchmarsh Eigenfunction expansions associated with second order differential equations Oxford University Press, Oxford, 1962.

18 Our approach: sector-dependent expansion We use an idea by L.Gr. Ixaru: L.Gr. Ixaru, New Numerical method for the eigenvalue problem of the 2D Schrödinger equation Computer Physics Communications 181 (2010) x [x min, x max ] t [0, T ] Inspired by Ixaru, [0, T ] is divided into sectors.

19 Our approach: sector-dependent expansion Sector k : t [t k 1, t k ] In sector k we approximate V (x, t) by V [k] (x), such that 2 Ĥ(x, t) = 1 + V (x, t) 2µ x 2 is approximated by d 2 Ĥ [k] (x) = 1 2µ dx 2 + V [k] (x)

20 Sectorwise approximation of V (x, t) In [t k 1, t k ] we approximate V (x, t) by V [k] (x) = V (x, (t k 1 + t k )/2) t x t x V (x, t) and V (x, t)

21 Propagation of the solution over one sector Assume ψ(x, t k 1 ) is known In [t k 1, t k ] we approximate the solution ψ(x, t) of the TDSE by ψ [k] (x, t) = N m=1 c [k] m (t)y [k] m (x)

22 Step 1 : choice of {y [k] m (x) m N} Let {y [k] m (x) m N} be the set of orthogonal eigenfunctions of Ĥ[k] (x). This requires the solution of the TISE Ĥ [k] (x)y [k] m (x) = E [k] m y [k] m (x)

23 ı d dt c[k] n (t) = Step 2: Computation of c [k] m (t) Substitute ψ [k] (x, t) = m=1 c [k] m (t)y [k] m (x) ψ(x, t) in the TDSE ı = Ĥ(x, t)ψ(x, t). t m=1 V nm(t) [k] xmax = x min [ V [k] nm(t) + E [k] m δ nm ] c [k] m (t), n = 1, 2, 3,... [ y [k] n (x) V (x, t) V ] [k] (x) y [k] m (x)dx

24 Step 2: Computation of c [k] m (t) Initial values are obtained imposing continuity at t k 1 : ψ [k 1] (x, t k 1 ) = ψ [k] (x, t k 1 ) m=1 c [k 1] m (t k 1 )y [k 1] m (x) = m=1 c [k] m (t k 1 )y [k] m (x). Multiply by y [k] n (x), integrate over x, and use orthonormality in the r.h.s. to obtain c [k] n (t k 1 ) = m=1 s [k] nmc [k 1] m (t k 1 ) s [k] nm = x max x min y [k] n (x)y [k 1] m (x)dx

25 Conclusion: to solve the TDSE in [t k 1, t k ] 1. solve the TISE Ĥ[k] (x)y [k] (x) = E [k] y [k] (x) 2. compute integrals s [k] nm = V nm(t) [k] xmax = x min x max x min y [k] n (x)y [k 1] m (x)dx [ y [k] n (x) V (x, t) V ] [k] (x) y [k] m (x)dx 3. solve a linear system C [k] (t) = A [k] (t)c [k] (t) initial conditions : C [k] (t k 1 ) = S [k] C [k 1] (t k 1 ) [ ] A [k] nm(t) = ı V nm(t) [k] + E [k] n δ nm

26 1. Solution TISE In order to have a computationially efficient approximation technique for the solution of the TISE, and to obtain a uniform high accuracy approximation to the eigenvalues and eigenfunctions we use Constant Perbutation methods as in MATSLISE). Here we use CP methods with a predefined uniform mesh (in order to minimize the number of CP eigenfunction evaluations).

27 2. Computation of the integrals Instead of using standard quadrature rules for integrands with a highly oscillatory or exponential character we use so-called exponentially-fitted quadrature rules especially adapted to this particular problem: the CP approach does not only provide us with the values of the wavefunctions at the meshpoints but also those of their first derivative. Ixaru L. Gr. Ixaru, and G. Vanden Berghe, Exponential Fitting, Kluwer, 2004.

28 Computation of s [k] nm = x max x min y [k] n Suppose one approximates y (x) = 2 µ(v (x) E) y(x) by ȳ (x) = 2 µ( V E)ȳ(x). Since (x)y [k 1] m (x)dx ȳ(x) = f 1 exp(2 µ V Ex) + f2 exp( 2 µ V Ex) one may expect y(x) = f 1 (x) exp(2 µ V Ex) + f2 (x) exp( 2 µ V Ex).

29 Computation of s [k] nm = x max x min y [k] n (x)y [k 1] m (x)dx If y [k] n (x) = f 1 (x) exp(ω [k] n x) + f 2 (x) exp( ω [k] n x) m (x) = g 1 (x) exp(ω [k 1] m x) + g 2 (x) exp( ω [k 1] m x) y [k 1] then y [k] n (x)y [k 1] m (x) = h 1 (x) exp(µ 1 x) + h 2 (x) exp( µ 1 x) +h 3 (x) exp(µ 2 x) + h 4 (x) exp( µ 2 x) µ 1 = ω [k] n + ω [k 1] m and µ 2 = ω [k] n ω [k 1] m

30 Computation of s [k] nm = x max x min y [k] n (x)y [k 1] m (x)dx We choose to use a 4-point Lobatto-EF algorithm of the form X+h X h I(x)dx h 4 n=1 a (0) n I(X + x n h) + h 2 4 n=1 which is exact for the functions a (1) n I (X + x n h) exp(±µ 1 x), exp(±µ 2 x), x exp(±µ 1 x), x exp(±µ 2 x).

31 xmax V nm(t) [k] = x min Computation of y [k] n (x) [ V (x, t) V [k] (x) ] y [k] m (x)dx If V (x, t) is explicitly known and the first derivative w.r.t. x of V (x, t) V [k] (x) can be evaluated, then the same rule can be applied. If this is not the case, then we use a Lobatto-type rule X+h X h I(x)dx h 4 n=1 a (0) n I(X + x n h) which is exact for the functions exp(±µ 1 x), exp(±µ 2 x).

32 3. Solving linear system of ODEs C [k] (t) = A [k] (t)c [k] (t) [ ] A [k] nm(t) = ı V nm(t) [k] + E [k] n δ nm initial conditions : C [k] (t k 1 ) = S [k] C [k 1] (t k 1 ) Idea : use coefficient approximation To obtain a second order method: replace A [k] (t) in [t k 1, t k ] by Ā[k] = A [k] ((t k 1 + t k )/2). In that case Ā[k] is diagonal such that C [k] (t) = exp[(t t k 1 )Ā[k] ]C [k] (t k 1 ), t [t k 1, t k ].

33 3. Solving linear system of ODEs To obtain higher order methods, we use similar techniques as in MATSLISE (perturbation theory and approximation of the coefficient functions by polynomials) C [k] (t) = DT D (t t k 1 )D T C [k] (t k 1 ) with T D (δ) = exp(δa D 0 ) + P 1(δ) + P 2 (δ) +... This is equivalent to using modified Neumann integral series. We have used a fourth order method with T D (δ) = exp(δa D 0 ) + P 1(δ)

34 3. Solving linear system of ODEs We use this fourth order scheme and applied it once over the full sector-length [t k, t k+1 ] or (if needed to ensure accuracy) on a subdivision of the sector [t k, t k+1 ].

35 3. Solving linear system of ODEs Remark: Only second order method is unitary, such that the norm is preserved over time. For the higher order methods, this is not the case. But: the correction terms are small and loss of norm conservation is consequently also likely to be small.

36 Example 1 V (x, t) = x 2 /2 2t x [ 10, 10], t [0, T ] = [0, 20] µ = 1 ψ(x, 0) = (2 n n! π) 1/2 exp( x 2 /2)H n (x) H n :Hermite polynomial of degree n ψ(x, t) = ψ(x, 0) exp( ı(n )t + ıt2 )

37 Example 1: t [0, 5] Re ψ(x, t) Im ψ(x, t) t x t x 10 x = 0.5, t = 0.25, N = 12 n = 2

38 Example 1: t [20, 21] Re ψ(x, t) Im ψ(x, t) t x t x 0 10 x = 0.5, t = 0.25, N = 12 n = 2

39 Error err N = x max ψ [K ] (x, T )ψ [K ] (x, T )dx x max ψ exact (x, T )ψ exact(x, T )dx x min x min err A = max ψ(x, T ) ψ exact (x, T ) x mesh x

40 Example 1 N = 10, K = 10, T = 20 N = 12, K = 5, T = 20 x t err N n = 2 7e-5 1e-10 8e-15 6e-5 6e-11 4e-14 n = 4 1e-3 3e-10 2e-14 7e-4 1e-10 1e-14 n = 6 1e-3 5e-10 2e-14 9e-4 2e-10 6e-14 err A n = 2 1e-4 5e-11 7e-14 5e-4 6e-11 5e-14 n = 4 5e-4 1e-10 6e-14 5e-4 2e-10 7e-14 n = 6 8e-4 1e-10 3e-14 6e-4 1e-10 5e-14 cputime (sec)

41 Example 1 Crank-Nicholson T = 2 T = 5 x t err N n = 2 4e-4 1e-4 5e-3 n = 4 2e-3 5e-4 1e-3 n = 6 4e-3 9e-4 3e-3 err A n = 2 3e-4 8e-5 8e-3 n = n = 6 8e-1 8e-1 6e-1 cputime (sec)

42 Example 2 V (x, t) = (4 3 exp( t)) x 2 /2 x [ 10, 10], t [0, T ] = [0, 12] µ = 1 ψ 0 (x) = ( ) 1 1/4 exp( 1 π 2 (x 2) 2 )

43 Example 2 N = 10, K = 20 N = 15, K = 20 x t err N 2e-3 5e-5 3e-9 2e-3 5e-5 6e-9 6e-11 err A 3e-3 8e-5 1e-5 3e-3 8e-5 3e-7 3e-7 time N = 20, K = 20 x t err N 6e-6 6e-9 6e-11 err A 8e-5 6e-8 2e-9 time

44 Example 2 x = 0.2 and t = 0.02 N K = 1 K = 10 K = e-2 3e-3 2e e-4 3e-5 9e e-5 3e-6 3e e-7 4e-8 2e e-9 3e-11 1e-12

45 Example 3 diatomic molecule in strong laser field (Walker and Preston) V (x, t) = D(1 e αx ) 2 +A cos(ωt)x x [ 1, 4.32] t [0, 100τ] τ = 2π/ω ω = ψ(x, 0) = ψ 0 (x) = σ exp( (ρ 1/2)αx) exp( ρe αx ) µ = 1745 D = α = A = (a.u.) ρ = 2D/ω 0, ω 0 = α 2D/µ and σ =

46 Example 3 x = x max x min 64 N = 15 K = 20 t τ/10 τ/20 τ/50 τ/100 τ/200 τ/400 err N 3e-3 1e-4 1e-6 2e-9 5e-10 5e-10 err A 1e-2 7e-4 2e-5 1e-6 8e-8 5e-9 time

47 Conclusions We illustrated a new method to solve the TDSE. CP technique can be used in the spatial discretization of a time-dependent Schrödinger equation for efficient and accurate time stepping in the resulting ODE system